My UV layout is distorted compared to the geometry of my object
When you unwrap an object and then add an image, the UV unwrapping will be adjusted to the aspect ratio of the image. So if you unwrap in the default square UV area and then open a rectangular image your UVs will be stretched/squashed.

Blender has an option when unwrapping called 'Correct Aspect' which should fix your exact problem. 'Correct Aspect' looks at the selected image node in the object's material and uses that to remove this stretching/squashing effect. However - it doesn't seem to like the ngon at the centre of that circle you are unwrapping.

So:


Select the ngon and press Ctrl+F> Poke Faces to triangulate it.
Press U to unwrap and, immediately after unwrapping, open the toolbar (T) in the 3D view and turn on 'Correct Aspect' or press F6 to access the same options.